Training Multi class classification (One-vs-all) on Neural network
Let us suppose that you are training a neural network for classfing images of vehicles , then the input vector , image of the "vehicle" will be a 2D array of pixels. This undergoes several transformations at each layer of the neural network , the last layer of the neural network produces another vector whose dimensions are lesser than the original image vector.

So the network is mapping images to some vectors in a high dimensional space. In order to classify the images it is now sufficient to classify the vectors obtained from the network of their corresponding images, you  can do this with a simple "linear" classifier using a softmax layer.

So all the layers of the network except the last layer are transforming the image representation to a "vector". This vector is classified by a linear softmax classifier by the last layer of neural network.